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Theorem fullfunfnv 31730
Description: The full functional part of 𝐹 is a function over V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fullfunfnv FullFun𝐹 Fn V

Proof of Theorem fullfunfnv
StepHypRef Expression
1 funpartfun 31727 . . . . 5 Fun Funpart𝐹
2 funfn 5882 . . . . 5 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
31, 2mpbi 220 . . . 4 Funpart𝐹 Fn dom Funpart𝐹
4 0ex 4755 . . . . . 6 ∅ ∈ V
54fconst 6053 . . . . 5 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
6 ffn 6007 . . . . 5 (((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} → ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
75, 6ax-mp 5 . . . 4 ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)
83, 7pm3.2i 471 . . 3 (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
9 disjdif 4017 . . 3 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
10 fnun 5960 . . 3 (((Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)) ∧ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅) → (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
118, 9, 10mp2an 707 . 2 (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
12 df-fullfun 31658 . . . 4 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
1312fneq1i 5948 . . 3 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V)
14 unvdif 4019 . . . . 5 (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V
1514eqcomi 2630 . . . 4 V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
1615fneq2i 5949 . . 3 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1713, 16bitri 264 . 2 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1811, 17mpbir 221 1 FullFun𝐹 Fn V
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  Vcvv 3189  cdif 3556  cun 3557  cin 3558  c0 3896  {csn 4153   × cxp 5077  dom cdm 5079  Fun wfun 5846   Fn wfn 5847  wf 5848  Funpartcfunpart 31632  FullFuncfullfn 31633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-symdif 3827  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31637  df-singleton 31645  df-singles 31646  df-image 31647  df-funpart 31657  df-fullfun 31658
This theorem is referenced by:  brfullfun  31732
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